Properties

Degree 4
Conductor $ 3^{4} \cdot 5^{4} \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·7-s + 2·13-s − 3·16-s + 3·19-s − 2·28-s + 10·31-s − 4·37-s − 4·43-s − 2·49-s − 2·52-s − 8·61-s + 7·64-s + 2·67-s − 4·73-s − 3·76-s + 4·79-s + 4·91-s + 14·97-s + 8·103-s − 14·109-s − 6·112-s + 14·121-s − 10·124-s + 127-s + 131-s + 6·133-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.755·7-s + 0.554·13-s − 3/4·16-s + 0.688·19-s − 0.377·28-s + 1.79·31-s − 0.657·37-s − 0.609·43-s − 2/7·49-s − 0.277·52-s − 1.02·61-s + 7/8·64-s + 0.244·67-s − 0.468·73-s − 0.344·76-s + 0.450·79-s + 0.419·91-s + 1.42·97-s + 0.788·103-s − 1.34·109-s − 0.566·112-s + 1.27·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.520·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(961875\)    =    \(3^{4} \cdot 5^{4} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{961875} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 961875,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.879716818$
$L(\frac12)$  $\approx$  $1.879716818$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.191063267747204379246728812215, −7.79299109407548105183651796404, −7.34081177954816747224363878474, −6.75495896163846866268716243282, −6.44506554897564806369745191099, −5.89865135636990142636287836411, −5.35321516780934268651586553298, −4.90169586335377487963386503725, −4.52740816792627750380325032619, −4.11883635151693151676754510789, −3.37280973143547339449146795377, −2.96647391623180758430622930866, −2.14148034080307729172786887827, −1.52644445037254599624864137489, −0.67335071423937653134473085898, 0.67335071423937653134473085898, 1.52644445037254599624864137489, 2.14148034080307729172786887827, 2.96647391623180758430622930866, 3.37280973143547339449146795377, 4.11883635151693151676754510789, 4.52740816792627750380325032619, 4.90169586335377487963386503725, 5.35321516780934268651586553298, 5.89865135636990142636287836411, 6.44506554897564806369745191099, 6.75495896163846866268716243282, 7.34081177954816747224363878474, 7.79299109407548105183651796404, 8.191063267747204379246728812215

Graph of the $Z$-function along the critical line